On consecutive 0 digits in the β-expansion of 1

Abstract For each 1 β 2 , let r n ( β ) be the maximal length of consecutive 0 digits in the first n digits of 1's β -expansions. We prove that for Lebesgue almost all 1 β 2 , lim n → ∞ ⁡ r n ( β ) / log β ⁡ n = 1 and r n ( β ) ≥ log β ⁡ n for infinitely many n , which answers two questions raised by Erdos, Joo and Komornik.